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Simplifying 0.25n(6n + -3) = 4n + 1 Reorder the terms: 0.25n(-3 + 6n) = 4n + 1 (-3 * 0.25n + 6n * 0.25n) = 4n + 1 (-0.75n + 1.5n2) = 4n + 1 Reorder the terms: -0.75n + 1.5n2 = 1 + 4n Solving -0.75n + 1.5n2 = 1 + 4n Solving for variable 'n'. Reorder the terms: -1 + -0.75n + -4n + 1.5n2 = 1 + 4n + -1 + -4n Combine like terms: -0.75n + -4n = -4.75n -1 + -4.75n + 1.5n2 = 1 + 4n + -1 + -4n Reorder the terms: -1 + -4.75n + 1.5n2 = 1 + -1 + 4n + -4n Combine like terms: 1 + -1 = 0 -1 + -4.75n + 1.5n2 = 0 + 4n + -4n -1 + -4.75n + 1.5n2 = 4n + -4n Combine like terms: 4n + -4n = 0 -1 + -4.75n + 1.5n2 = 0 Begin completing the square. Divide all terms by 1.5 the coefficient of the squared term: Divide each side by '1.5'. -0.6666666667 + -3.166666667n + n2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -3.166666667n + 0.6666666667 + n2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -3.166666667n + n2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -3.166666667n + n2 = 0 + 0.6666666667 -3.166666667n + n2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -3.166666667n + n2 = 0.6666666667 The n term is -3.166666667n. Take half its coefficient (-1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. -3.166666667n + 2.506944447 + n2 = 0.6666666667 + 2.506944447 Reorder the terms: 2.506944447 + -3.166666667n + n2 = 0.6666666667 + 2.506944447 Combine like terms: 0.6666666667 + 2.506944447 = 3.1736111137 2.506944447 + -3.166666667n + n2 = 3.1736111137 Factor a perfect square on the left side: (n + -1.583333334)(n + -1.583333334) = 3.1736111137 Calculate the square root of the right side: 1.781463195 Break this problem into two subproblems by setting (n + -1.583333334) equal to 1.781463195 and -1.781463195.Subproblem 1
n + -1.583333334 = 1.781463195 Simplifying n + -1.583333334 = 1.781463195 Reorder the terms: -1.583333334 + n = 1.781463195 Solving -1.583333334 + n = 1.781463195 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + n = 1.781463195 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + n = 1.781463195 + 1.583333334 n = 1.781463195 + 1.583333334 Combine like terms: 1.781463195 + 1.583333334 = 3.364796529 n = 3.364796529 Simplifying n = 3.364796529Subproblem 2
n + -1.583333334 = -1.781463195 Simplifying n + -1.583333334 = -1.781463195 Reorder the terms: -1.583333334 + n = -1.781463195 Solving -1.583333334 + n = -1.781463195 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + n = -1.781463195 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + n = -1.781463195 + 1.583333334 n = -1.781463195 + 1.583333334 Combine like terms: -1.781463195 + 1.583333334 = -0.198129861 n = -0.198129861 Simplifying n = -0.198129861Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.364796529, -0.198129861}
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